$\dfrac{2n+1} { [n(n+1)]^2}=\dfrac{n}{[n(n+1)]^2}+\dfrac{n+1}{[n(n+1)]^2}=\dfrac{1}{n(n+1)^2}+\dfrac{1}{n^2(n+1)}=\dfrac{1}{n(n+1)}(\dfrac{1}{n+1}+\dfrac{1}{n})=(\dfrac{1}{n}-\dfrac{1}{n+1})(\dfrac{1}{n}+\dfrac{1}{n+1})= $
$=\dfrac{1}{n^2}-\dfrac{1}{(n+1)^2}$
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